Question

Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.) Suppose \(a, b \in \mathbb{R} .\) If \(a\) is rational and \(a b\) is irrational, then \(b\) is irrational.

Short answer

From this proof by contradiction, we can say that if \(a\) is rational and \(ab\) is irrational, then \(b\) must be irrational. This proves the stated claim to be true.

Step-by-step solution

Define Rational and Irrational Numbers

Before beginning the proof, establish the definitions of rational and irrational numbers. A rational number can be expressed as the quotient of two integers (with the denominator not equal to 0), while an irrational number cannot be expressed as a simple fraction, and hence is not rational.

State Assumption for Proof by Contradiction

In a proof by contradiction or indirect proof, begin by assuming the opposite of what is to be proven. That is, assume \(a\) is rational, \(ab\) is irrational, and \(b\) is rational.

Apply Rules of Rational Numbers

Since both \(a\) and \(b\) are assumed to be rational, and the product of two rational numbers is always rational, therefore \(ab\) must be rational. Now we have two contradictory statements: \(ab\) is both irrational (from our initial assumption) and rational (from the rules of rational numbers).

Draw the Contradiction and Conclude

This contradiction indicates that our assumption was incorrect. Therefore, if \(a\) is rational and \(ab\) is irrational, \(b\) must be irrational.